Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

Abstract

We study the dynamics of the following porous medium equation with strong absorption [Formula: see text] posed for [Formula: see text], with [Formula: see text], [Formula: see text] and [Formula: see text]. Considering the Cauchy problem with non-negative initial condition [Formula: see text], instantaneous shrinking and localization of supports for the solution [Formula: see text] at any [Formula: see text] are established. With the help of this property, existence and uniqueness of a non-negative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.